ISIMA lectures on celestial mechanics. 3

Scott Tremaine, Institute for Advanced Study

July 2014

1. The stability of planetary systems

To understand the formation and evolution of exoplanet systems, we would like to have

empirical theoretical tools for predicting whether a given planetary system is stable—or,

more precisely, what is its lifetime before some catastrophic event such as a collision or

ejection—without having to integrate the planetary orbits for the lifetime of the host star.

There are two discouraging lessons from studies of the stability of the solar system that we

should bear in mind. First, small changes in the initial conditions or system parameters can

lead to large changes in the lifetime. Second, chaos does not necessarily imply instability:

in the solar system the likely lifetime is at least 1000 times longer than the Liapunov time.

One approach to this problem is to consider planetary systems in which both the planet

mass and the separation between adjacent planets are very small. In examining this limit

a useful parameter is the Hill radius. Consider a planet of mass m in a circular orbit of

radius r around a host star of mass M . A test particle also circles the host star, in an orbit

of radius r + x with 0 < x r. The planet orbits with angular speed n = (GM/r3)1/2

which is slightly faster than the angular speed of the test particle, [GM/(r+ x)3]1/2. As the

planet overtakes the test particle the maximum force per unit mass that it exerts on the

test particle is F ∼ Gm/x2 and this force lasts for a duration ∆t ∼ n−1; thus the velocity

change is ∼ F∆t ∼ Gm/(x2n). The change is as large as the initial relative velocity v ∼ nx

if F∆t & v which implies x3 . Gm/n2 = r3(m/M), or x . rH where

rH = r( m

3M

)1/3(1)

is the Hill radius. The Hill radius appears in other contexts in astrophysics, such as the tidal

or limiting radius of star clusters, the Roche limit for solid satellites, and the location of the

Lagrange points (the factor of 3 in this formula ensures that the Hill radius is equal to the

distance to the collinear Lagrange points).

For two planets of comparable mass, m1,m2 M , the analog is the mutual Hill

radius, defined as

rH =r1 + r2

2

(m1 +m2

3M

)1/3

. (2)

– 2 –

The use of the arithmetic mean of the radii (rather than, say, the geometric mean) is arbitrary

but the choice of what mean to use makes very little difference since the mutual Hill radius is

only dynamically important when r1 and r2 are similar, since we have assumed thatm1,m2 M . In contrast, we use the sum of the masses (rather than, say, (m1/3M)1/3 + (m2/3M)1/3)

because the dynamics of two small bodies during a close encounter depends on the masses

only through their sum (Petit & Henon 1986, Icarus 66, 536).

A system of two small planets on nearby circular, coplanar orbits is stable for all time if

|r1− r2| > 3.46rH (the standard reference is Gladman 1993, Icarus 106, 247, although forms

of this criterion were derived earlier). Note that the scaling |r1 − r2| > const × (m/M)1/3

does not capture all of the interesting dynamics of two bodies on nearby orbits; for example,

the criterion that the orbits are regular rather than chaotic is |r1 − r2| > const× (m/M)2/7

(Wisdom 1980, AJ 85, 1122).

For systems of three or more planets there are no exact stability criteria. Moreover such

systems are not stable for all time—the instability time becomes very large if the planets

have small masses or large separations but such systems always eventually exhibit instability.

Numerical orbit integrations imply that a system of three or more planets on nearly circular,

coplanar orbits is stable for at least N orbital periods if the separation between adjacent

planets is |ri+1− ri| > K(N)rH where, for example, K(1010) ' 9–12 (e.g., Smith & Lissauer

2009, Icarus 201, 381).

In summary, a system of planets on nearly circular, coplanar orbits is expected to be

stable for the stellar lifetime if the separation of adjacent planets exceeds ∼ 10 mutual Hill

radii. To do better than this crude criterion, our only option is to integrate the planetary

orbits.

2. Hamiltonian perturbation theory

2.1. The disturbing function

The dynamics of any one body in the N-body problem can be described by a Hamiltonian

H(q,p) = HK(q,p)+H1(q,p, t) where HK = 12|p|2−GM/|q| is the Kepler Hamiltonian and

H1 represents the perturbing gravitational potential from the other planets, or from passing

stars, the equatorial bulge of the host star, etc. We are interested mainly in near-Keplerian

problems where |H1| |HK |. The goal of perturbation theory in celestial mechanics is to

find solutions for motion in this Hamiltonian that are valid when H1 is “small enough”—in

fact it is for this problem that perturbation theory was invented. The goal of this section is

to persuade you that Hamiltonian perturbation theory is straightforward to apply in simple

– 3 –

systems, and provides answers and insight that cannot easily be obtained in any other way.

In a Hamiltonian system we can replace (q,p) with any other canonical coordinates and

momenta. In particular we may use the Delaunay elements1 to obtain

H = −(GM)2

2L2+H1(L,G,H, `, ω,Ω, t). (3)

The function H1 is called the disturbing function (unfortunately in many celestial me-

chanics books this name is used to refer to −H1). Deriving H1 is much harder in Delaunay

elements than in Cartesian coordinates (for example, it depends on all six Delaunay elements

but only three Cartesian coordinates) but the advantage of working in Delaunay elements is

so great that this effort is usually worthwhile.

Some restrictions on the form of H1 can be derived from general considerations. If we

replace ` by ` + 2π the position of the particle is unchanged, so H1 is unchanged. Thus H1

must be a periodic function of ` with period 2π. Similarly it must be a periodic function of

ω and Ω with the same period. Thus it can be written as a Fourier series

H1 =∑k,m,n

Hkmn(L,G,H, t) cos[k`+mω + nΩ + φ`mn(t)]

=∑k,m,n

Hkmn(L,G,H, t) cos[kλ+ (m− k)$ + (n−m)Ω + φkmn(t)], (4)

where as usual the mean longitude λ = `+ω+ Ω and the longitude of periapsis $ = ω+ Ω.

When the eccentricity is zero (G = L), H1 cannot depend on the periapsis direction $.

Similarly, when the inclination is zero (H = G), H1 cannot depend on the nodal longitude

Ω. More generally, H1 has to be an analytic function of the position and this restricts the

possible forms of H1 to satisfy the d’Alembert property2

Hkmn(L,G,H, t) = O[(L−G)|m−k|/2, (G−H)|n−m|/2]. (5)

Now suppose that the perturbing potential is stationary in a rotating frame, Φ[R, z, φ −φs(t)] =

∑m′ Φm′(R, z) cosm′[φ− φs(t)]. If we rotate the origin of the azimuthal coordinate

system by ψ then φ and φs both increase by ψ but the potential remains the same. Similarly,

1As a reminder, the Delaunay momenta are L = (GMa)1/2, G = L(1 − e2)1/2, H = G cos i and the

conjugate coordinates are the mean anomaly `, the argument of periapsis ω, and the longitude of the

ascending node Ω.

2Similarly, if a smooth function f(x, y) of the Cartesian coordinates x and y is rewritten in polar coordi-

nates (r, φ) then its expansion in powers of r will have the form c0 +c1r cos(φ−φ1)+c2r2 cos 2(φ−φ2)+ · · · .

– 4 –

the longitude of the node Ω increases by ψ but the elements ` and ω do not change. In order

that H1 does not change, we then require that m = m′ and φkmn = −mφs(t).

Thousands of pages have been written about the disturbing function in the gravitational

N-body problem and its approximate forms for nearly circular, nearly coplanar orbits like

those found in the solar system. Here we will discuss only three specific problems to give

some flavor of these results and how the disturbing function is used in practice.

2.2. Satellite orbits

We first analyze some of the properties of the orbit of a satellite around a planet. The

external gravitational potential from a planet with an axis of symmetry can be written

Φ(r) = −Gmr

[1−

∞∑n=2

JnRn

rnPn(cos θ)

](6)

where m and R are the mass and mean radius of the planet, θ is the polar angle from the

symmetry axis, Pn is a Legendre polynomial, and Jn is a dimensionless multipole moment.

There is no n = 1 component if the center of mass is chosen to be the origin of the coordinates.

In most cases the dominant component is n = 2; since P2(cos θ) = 32

cos2 θ− 12

we then have

Φ(r) = −Gmr

+GmJ2R

2

r5(3z2 − r2) (7)

where z is the component of the radius vector along the symmetry axis. The corresponding

disturbing function is

H1 =GmJ2R

2

2r5(3z2 − r2). (8)

In converting this potential to a disturbing function of the form (4) we note that Φ

has no explicit time dependence so φkmn is independent of time. Since the mean longitude

λ rotates through 2π in one orbital period, while $ and Ω are constants for a Keplerian

orbit, the terms in the disturbing function are of two types: short-period terms with k 6= 0

and secular terms with k = 0. Although the short-period terms are important for precise

predictions of the location of the satellite at a given instant, the largest changes in its orbit

will come from the secular terms since these act for a much longer time.

Restricting the disturbing function to its secular terms is equivalent to averaging over

one orbit. Denoting this average by 〈·〉 and evaluating 〈z2/r5〉 and 〈1/r3〉 using equations

(50) and (43) of Lecture 1, we find

Hsec = 〈H1〉 =GmJ2R

2

4a3(1− e2)3/2(3 sin2 I − 2) =

(Gm)4J2R2

4L3G3(1− 3H2/G2). (9)

– 5 –

With this disturbing function, most of the properties of the satellite orbit are easy to

understand. For example, Hamilton’s equations imply that dL/dt = −∂Hsec/∂` = 0 so

the semi-major axis a = L2/(GM) is constant, except for small short-period variations.

Similarly, dG/dt = −∂Hsec/∂ω = 0 so the angular momentum and eccentricity are constants

except for short-period variations3. Similarly dH/dt = −∂Hsec/∂Ω = 0 so H is constant and

the inclination I = cos−1H/G is fixed. The apsis direction or perigee ω precesses at a rate

ω =∂Hsec

∂G=

3(Gm)4J2R2

4L3G6(5H2 −G2)

=3(Gm)4J2R

2

4L3G6(5H2 −G2)

=3(Gm)1/2J2R

2

4a7/2(1− e2)2(5 cos2 I − 1). (10)

Thus the apsis direction of low-inclination orbits precesses forward; the apsis direction of

high-inclination orbits precesses backward; and at the so-called critical inclination I =

cos−1√

1/5 = 63.4 the apsis does not precess.

The node precesses at a rate

Ω =∂Hsec

∂H= −3(Gm)1/2J2R

2

2a7/2(1− e2)2cos I, (11)

which is retrograde for all orbits with 0 ≤ I < 90.

2.3. Lidov-Kozai oscillations

Suppose that a planet of mass m orbits a host star of mass m0, with m + m0 = M . Its

position relative to the host star is x. There is a companion star of mass mc with position

xc relative to the host star, with |xc| |x|. Expanding the heliocentric potential (eq. 37 of

Lecture 2) in inverse powers of |xc|,

Φ(x) = −GMx− Gmc

xc√

1− 2x · xc/x2c + x2/x2c+Gmcx · xc

x3c

= −GMx− Gmc

x3c

[x2c + x · xc − 1

2x2 +

3(x · xc)2

2x2c

]+Gmcx · xc

x3c+ O(x3/x4c)

= −GMx

+Gmc

x3c

[12x2 − 3(x · xc)

2

2x2c

]+ O(x3/x4c); (12)

3This is a property of the quadrupole potential that is not shared by other multipoles.

– 6 –

in the last line we have dropped an unimportant constant term −Gmc/xc. Dropping the

terms of order x3/x4c gives us the quadrupole approximation, in which the disturbing

function is

H1 =Gmc

x3c

[12x2 − 3(x · xc)

2

2x2c

]. (13)

Let e, u, n be unit vectors pointing respectively towards periapsis (f = 0), towards f = 12π

in the orbital plane, and along the angular-momentum vector (thus e × u = n). Then

x = r(f)(cos f e + sin f u) with a similar formula for xc. As in the previous section, the

largest effect on the orbit comes from the secular terms, which are obtained by averaging H1

over the orbit. In this case we want to average over both the inner and outer orbits:

Hsec = 〈H1〉 =Gmc

[12〈r2〉

⟨1

r3c

⟩− 3

2(e · ec)2〈r2 cos2 f〉

⟨cos2 fcr3c

⟩− 3

2(e · uc)

2〈r2 cos2 f〉⟨

sin2 fcr3c

⟩− 3

2(u · ec)2〈r2 sin2 f〉

⟨cos2 fcr3c

⟩− 3

2(u · uc)

2〈r2 sin2 f〉⟨

sin2 fcr3c

⟩]. (14)

Using equations (43) of Lecture 1, this simplifies to

Hsec =Gmca

2

4a3c(1− e2c)3/2

2 + 3e2 − 32[(e · ec)2 + (e · uc)

2](1 + 4e2)

− 32[(u · ec)2 + (u · uc)

2](1− e2)

(15)

Using the relation (e · ec)2 +(e · uc)2 +(e · nc)

2 = 1 and a similar relation for u, this simplifies

further to

Hsec =Gmca

2

4a3c(1− e2c)3/2[−1− 3

2e2 + 3

2(e · nc)

2(1 + 4e2) + 32(u · nc)

2(1− e2)]. (16)

We choose the reference frame to be the orbital plane of the companion star. Then

e · nc = sin I sinω and u · nc = sin I cosω and

Hsec =Gmca

2

8a3c(1− e2c)3/2[1− 6e2 − 3(1− e2) cos2 I + 15e2 sin2 I sin2 ω

]=

mcL4

8GM2a3c(1− e2c)3/2

[6G2

L2− 5− 3H2

L2+ 15

(1− G2

L2

)(1− H2

G2

)sin2 ω

]. (17)

The secular Hamiltonian is independent of ` so the semi-major axis is constant, and indepen-

dent of Ω so H is constant. In contrast to the Hamiltonian (9), however, this Hamiltonian

depends on the argument of periapsis ω, so the conjugate variableG (the angular momentum)

– 7 –

is not constant. This gives rise to the remarkable phenomenon of Lidov-Kozai oscillations,

which we now describe.

Suppose that the initial planetary orbit is circular and inclined by I0 to the reference

plane (the orbital plane of the binary companion). Then H = (GMa)1/2 cos I0 and H and

L are conserved, so conservation of Hsec requires

2− 2e2 + 5 sin2 ω(e2 − sin2 I0) = 0 or e = 0. (18)

It is not hard to show that the first of these equations has a solution for 0 < e < 1 if and

only if I0 > sin−1√

2/5 = 39.2. For larger initial inclinations the circular orbit may (and

does) undergo oscillations to high eccentricity (Lidov–Kozai oscillations). The maximum

eccentricity occurs at sinω = 1 and is equal to

emax =

√5 sin2 I0 − 2

3; (19)

as I0 approaches 90 the maximum eccentricity approaches unity (i.e., a collision orbit).

Contours of equation (18 for I0 = 60 are shown in Figure 1.

Among the remarkable properties of Lidov–Kozai oscillations are that (i) above the

critical inclination of 39.2, circular orbits are chaotic (because of additional terms due

to octupole and higher moments of the tidal potential); (ii) the mass and distance of the

companion star affect the period of the oscillations but not their amplitude. This means

that in principle arbitrarily small external forces can lead to large changes in the orbit. In

practice, when the perturber is sufficiently weak, either the oscillation period becomes longer

than the age of the system, or other sources of precession—additional planets or general

relativity—become stronger than the precession due to the companion, and this quenches

the oscillations.

Note that there is an accidental degeneracy in the problem as formulated here: the

Hamiltonian does not depend on the orientation of the apsis of the companion star. In more

realistic treatments that include higher-order multipole moments this degeneracy is lifted,

giving the dynamics an additional degree of freedom and thus more complicated behavior.

Lidov–Kozai oscillations are important in a remarkable variety of astrophysical contexts:

the irregular satellites of the giant planets (Nesvorny et al. 2003, AJ 126, 398); the excitation

of exoplanet eccentricities by companion stars (Holman et al. 1997, Nature 386, 254); the

formation of close binary stars (Fabrycky & Tremaine 2007, ApJ 669, 1298); blue straggler

stars (Perets & Fabrycky 2009, ApJ 697, 1048); Type Ia supernovae (Thompson 2011, ApJ

741, 82; Kushnir et al. 2013, ApJ 778, L37); planetary migration; black-hole mergers (Blaes

et al. 2002, ApJ 578, 775); asteroid orbits; satellite orbits; comets; etc.

– 8 –

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

1.0

Fig. 1.— Lidov–Kozai oscillations of a planet that is initially in a circular orbit with inclination

I0 = 60 to the orbit of a distant companion star. The vertical axis is eccentricity and the horizontal

axis is argument of periapsis. The contours are level surfaces of the energy (18).

2.4. Mean-motion resonances

Orbital resonances are ubiquitous in the solar system (Peale 1976, ARAA 14, 215). Some

examples include the 3:2 resonance between the orbital periods of Neptune and Pluto and

other Kuiper belt objects; the 2:1 resonance between the orbital periods of Saturn’s satellites

Mimas and Tethys, and Dione and Enceladus; the co-orbital (1:1) resonance between Saturn’s

satellites Janus and Epimetheus; the Kirkwood gaps in the asteroid belt and the gaps in

Saturn’s rings; the three-body resonance between the mean motions of the first three Galilean

satellites (nIo − 3nEuropa + 2nGanymede = 0); Trojan asteroids; etc.

To keep the discussion as simple as possible, we focus on resonances between a planet

on a circular orbit and a test particle orbiting on a near-circular orbit interior to the planet.

We shall also assume that the orbits are coplanar. We denote the semi-major axis and mean

motion of the planet’s orbit by ac and nc, and its mass by mc. According to the discussion

– 9 –

in §2.1 the disturbing function can be written

H1 =∑k,m,n

Hkmn(L,G,H) cos(k`+mω + nΩ−mnst); (20)

we have chosen the origin of time so that the planet is at azimuthal angle φc = 0 at t = 0.

The Hamiltonian is further restricted by the d’Alembert property (5). We are assuming

that the satellite and test-particle orbits are coplanar so G = H, which implies that the only

non-zero terms have n = m. We are also assuming that the eccentricity of the test particle

is small, so Hkmn ∼ e|m−k|. Thus the largest terms in the Hamiltonian have k = m, so we

may approximate

H1 =∑m

Am cos[m(`+ ω + Ω− nct)] =∑m

Hm(L) cos[m(λ− nct)]. (21)

Resonance occurs when the rate of change of the argument of the cosine is small, that is,

when n ' nc. In other words the mean motions of the test particle and the planet must be

nearly the same; examples of this resonance (also called a 1:1 resonance) include horseshoe

orbits and Trojan orbits. These are quite interesting but we will not discuss them further

because of time limitations.

The next strongest resonances occur when Hkmn ∼ e and require m = k ± 1. For these

H1 can be approximated as

H1 =∑m

√L−GAm cos[(m∓ 1)`+mω +mΩ−mnct]

=∑m

√L−GAm cos[(m∓ 1)λ±$ −mnct] (22)

where Am is a constant (in principle Am depends on L but since the variations in L are

small we can treat Am as a constant). Resonance occurs when (m∓ 1)n = mnc. There are

also resonances with (m ∓ j)nc = mnc with j > 1, but these are weaker since the resonant

Hamiltonian is ∼ ej by the d’Alembert property.

Since we have assumed that the test particle is interior to the planet, and since we can

restrict ourselves without loss of generality to m ≥ 0, we must choose the upper sign, so

n = mnc/(m − 1) or n/nc = 2, 32, 43, · · · . These are known as mean-motion or Lindblad

resonances.

We can simplify the problem by a canonical transformation using the generating function

S = −Js[(m− 1)`+mω +mΩ−mnct] + Jf1(`+ ω) + Jf2(`+ ω + Ω). (23)

– 10 –

The new coordinates are

θs =∂S

∂Js= −(m− 1)`−mω −mΩ +mnct, θf1 = `+ ω, θf2 = `+ ω + Ω. (24)

The subscript “s” indicates that θs varies slowly near the resonance while “f” indicates that

θf1 and θf2 vary rapidly (θf1 ' θf2 ' n). The new momenta are related to the Delaunay

momenta by

L =∂S

∂`= (1−m)Js + Jf1 + Jf2, G =

∂S

∂ω= −mJs + Jf1 + Jf2, H = −mJs + Jf2. (25)

Solving for the new momenta,

Js = L−G, Jf1 = G−H, Jf2 = H +m(L−G), (26)

and the new Hamiltonian is

H = HK +H1 +∂S

∂t= − (GM)2

2[(1−m)Js + Jf1 + Jf2]2+ Am

√Js cos θs +mncJs. (27)

The Hamiltonian is independent of the fast coordinates θf1 and θf2 so Jf1 and Jf2 are

constants. Since the inclination is zero G = H; thus Jf1 = 0. Moreover since the eccentricity

is small Js is small and Jf2 ' L, which is not small. Thus we can expand the denominator

in equation (27) to quadratic order in Js/Jf2. Dropping an unimportant constant term, we

have

H = Js[mnc − (m− 1)nr]−3(m− 1)2

2a2rJ2s + Am

√Js cos θs; (28)

here we have defined the constants ar and nr by Jf2 = (GMar)1/2 and nr = (GM/a3r)

1/2.

When the perturbing potential Am is small and the eccentricity is also small (so Js is small),

we have θs ' ∂H/∂Js = mnc− (m− 1)nr so this constant can be thought of as the distance

away from exact resonance, where θs = 0.

Rescaling the action and the Hamiltonian by constant factors allows us to convert the

Hamiltonian into a standard form (Henrard & Lemaıtre 1983, Cel. Mech. 30, 197; Borderies

& Goldreich 1984, Cel. Mech. 32, 127; Petrovich et al. 2013, ApJ 770, 24)

H = −3∆R +R2 − 2√

2R cos r, (29)

where R and r is a conjugate momentum-coordinate pair and ∆ ∝ mnc−(m−1)nr measures

the distance away from resonance. Henrard & Lemaıtre call this the “second fundamental

model for resonance” in contrast to the “first fundamental model for resonance” which is the

simple pendulum,

H = −∆R + 12R2 − cos r. (30)

– 11 –

The Hamiltonian (29) has only one degree of freedom and thus is integrable. Thorough

discussions of its solutions and how they evolve under slow changes in ∆ (as might be caused,

e.g., by migration or growth of the mass of the planet) are given in the references above.

They provide an analytical description of such phenomena as resonance capture, eccentricity

excitation during resonance crossing, etc. I shall not discuss these results due to lack of

time, and since they are covered in the references. The main purpose of this discussion

has been to show how perturbation theory can be used to derive a model Hamiltonian that

is simple enough to solve analytically but realistic enough to capture the main features of

mean-motion resonances.